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# Definite integral of radical function

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.B (LO)
,
FUN‑6.C (LO)

## Video transcript

so we want to evaluate the definite integral from negative 1 to 8 of 12 times the cube root of X DX let's see this is going to be the same thing it's the definite integral from negative 1 to 8 of 12 times the cube root is the same thing as saying X to the 1/3 power DX and so now if we want to take the antiderivative of this stuff on the inside we're just going to do essentially the power rule you could use as a power rule of integrals or it's the reverse of the power rule for derivatives where we increase this exponent by 1 and then we divide by that increased exponent so this is going to be equal to 12 times X to the 1/3 plus 1 let me don't let me do it another color just we can keep track of it X to the 1/3 plus 1 and then we're going to divide by 1/3 plus 1 and so what's 1/3 plus 1 well that's 4/3 1/3 plus 3/3 that's 4/3 so I could write it this way I could write this X to the 4/3 divided by 4/3 and this is going to be and I'm going to evaluate this at the bounds so I'm going to evaluate this at and I'll do this in different colors I'm going to evaluate it at 8 and I'm going to evaluate it at negative 1 and I'm going to I'm going to subtract it evaluated at negative 1 from this expression evaluated at 8 and so what is this going to be equal to well actually let me simplify a little bit more what is 12 divided by 4/3 so 12 I'll do it right well I'll do it right over here 12 over 4/3 is equal to 12 times 3 over 4 which we could view as 12 over 1 times 3 over 4 12 divided by 4 is 3 so this is going to be equal to 9 3 fourths of 12 is 9 so this we could rewrite this we could write this as 9x to the 4/3 power so if we evaluate it at 8 this is going to be 9 times 8 to the 4/3 power and from that we're going to subtract it evaluated at negative one so this is going to be nine times negative 1 to the 4/3 power so what is 8 to the 4/3 power I'll do it over here so 8 to the 4/3 is equal to 8 to the 1/3 to the 4th power these are just exponent properties here 8 to the 1/3 the cube root of 8 or 8 to the 1/3 power that's 2 because 2 to the third power is 8 and 2 to the 4th power well 2 to the 4th power is equal to 16 so 8 to the 4/3 is 16 and what's negative 1 to the 4/3 well same idea negative 1 to the 4/3 is equal to negative 1 there's several ways you can do it you can say negative 1 to the 4th and then the cube root of that or the cube root of negative 1 and then raise that to the 4th power either way so let's do it the first way negative 1 to the 4th and then take the cube root of that well negative 1 to the 4th is just 1 and then 1 to the 1/3 power well that's just going to be equal to 1 so what we have here in blue that's just equal to 1 so we have 9 times 16 minus 9 times 1 well that's just going to be 9 times 15 we have 16 9s and then we're going to take away a 9 so that's going to be 9 times 15 so what is that that is going to be equal to 9 times 15 is 90 plus 45 which is equal to 135 135 and we're done
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